Definition. Let PPP be a set and nnn a positiveinteger. An incidence geometry on PPP consists of

an onto function t:P→{0,1,…,n}normal-:tnormal-→P01normal-…nt\colon P\to\{0,1,\ldots,n\} called a type function. If we define Pi:=t-1⁢(i)assignsubscriptPisuperscriptt1iP_{i}:=t^{{-1}}(i), then PPP can be partitioned into a finitenumber of subsets:

P=P0∪⋯∪Pn, with ⁢Pi∩Pj=∅⁢ for ⁢i≠j.formulae-sequencePsubscriptP0normal-⋯subscriptPn with subscriptPisubscriptPj for ijP=P_{0}\cup\cdots\cup P_{n},\mbox{ with }P_{i}\cap P_{j}=\varnothing\mbox{ for%
}i\neq j.

Elements of PisubscriptPiP_{i} are variously known as blocks or varieties of typeiii. Sometimes, they are also called flats of dimensioniii. For this discussion, we will use the latter terminology. Flats of specific dimensions have further conditions:

(a)

P0≠∅subscriptP0P_{0}\neq\varnothing. Flats of dimension 0 are called points.

(b)

Pn≠∅subscriptPnP_{n}\neq\varnothing and consists of one element SSS, called the
space.

a reflexive and symmetric relationI⊆P×PIPPI\subseteq P\times P on PPP, called an incidence relation, with the following conditions (or axioms):

(a)

for any a,baba,b such that (a,b)∈IabI(a,b)\in I and t⁢(a)=t⁢(b)tatbt(a)=t(b), then a=baba=b;

(b)

suppose t⁢(a)≤t⁢(b)≤t⁢(c)tatbtct(a)\leq t(b)\leq t(c) with (a,b)∈IabI(a,b)\in I and (b,c)∈IbcI(b,c)\in I, then (a,c)∈IacI(a,c)\in I;

(c)

given that t⁢(a)<ntant(a)<n, there is a point ppp such that (p,a)∉IpaI(p,a)\notin I;

(d)

given t⁢(a)<ntant(a)<n and a point ppp with (p,a)∉IpaI(p,a)\notin I, there is a unique bbb with t⁢(b)=t⁢(a)+1tbta1t(b)=t(a)+1, such that (a,b)∈IabI(a,b)\in I and (p,b)∈IpbI(p,b)\in I; furthermore, if, in addition, there is a ccc such that (p,c)∈IpcI(p,c)\in I and (a,c)∈IacI(a,c)\in I, then (b,c)∈IbcI(b,c)\in I as well;

(e)

given that t⁢(a)>0ta0t(a)>0, there is a pair of a point ppp and a flat bbb, with t⁢(b)=t⁢(a)-1tbta1t(b)=t(a)-1 and (p,b)∉IpbI(p,b)\notin I, such that (b,a)∈IbaI(b,a)\in I and (p,a)∈IpaI(p,a)\in I;

(f)

given that 0<t⁢(a)=t⁢(b)=i<n0tatbin0<t(a)=t(b)=i<n, a point ppp with (p,a),(p,b)∈IpapbI(p,a),(p,b)\in I, and a flat ddd with t⁢(d)=i+1tdi1t(d)=i+1, (a,d),(b,d)∈IadbdI(a,d),(b,d)\in I, then there is a ccc with t⁢(c)=i-1tci1t(c)=i-1 such that (a,c),(b,c)∈IacbcI(a,c),(b,c)\in I.

An incidence geometry is often written as a triple (P,n,I)PnI(P,n,I).

Remarks.

Flats of dimensions 1 and 2 are commonly called lines and planes, respectively. Flats of dimension n-1n1n-1 are called hyperplanes.

When (a,b)∈IabI(a,b)\in I, we say that aaa is incident with bbb. Since III is symmetric, we also say that aaa and
bbb are incident without any ambiguity. Furthermore, if aaa and bbb are incident with t⁢(a)≤t⁢(b)tatbt(a)\leq t(b), then we say that aaalies onbbb, or bbbpasses throughaaa.

From Conditions 4 and 5 above, given any flat aaa of dimension i>0i0i>0, there exist a point ppp and a flat bbb of dimension i-1i1i-1 with the properties that

(a)

ppp is not incident with bbb,

(b)

aaa is incident with ppp, and

(c)

aaa is incident with bbb,

then the unique flat of dimension iii mentioned in Condition 4 is aaa. We say that aaa is generated byppp and bbb, or that ppp and bbbgenerateaaa, and we write a=⟨p,b⟩apba=\langle p,b\rangle. When bbb is a point, aaa is often written as p⁢b↔normal-↔pb\overleftrightarrow{pb}. In addition, if we were to pick a different pair of a point p′superscriptpnormal-′p^{{\prime}} and a flat b′superscriptbnormal-′b^{{\prime}} of dimension i-1i1i-1 satisfying the above three properties, then a=⟨p′,b′⟩asuperscriptpnormal-′superscriptbnormal-′a=\langle p^{{\prime}},b^{{\prime}}\rangle as well.

From the second part of Condition 4, ⟨p,b⟩pb\langle p,b\rangle is, in a sense, the smallest block (in terms of its type number), such that the above three properties hold. In other words, for any block aaa with (p,a)∈IpaI(p,a)\in I and (b,a)∈IbaI(b,a)\in I, then (⟨p,b⟩,a)∈IpbaI(\langle p,b\rangle,a)\in I. It is easy to see that t⁢(a)≥t⁢(⟨p,b⟩)tatpbt(a)\geq t(\langle p,b\rangle). For otherwise, t⁢(a)<t⁢(⟨p,b⟩)tatpbt(a)<t(\langle p,b\rangle), which meanst⁢(a)≤t⁢(b)tatbt(a)\leq t(b). This inequality together with (p,a)∈IpaI(p,a)\in I and (a,b)ab(a,b) imply
that (p,b)∈IpbI(p,b)\in I, a contradiction.

In Condition 6, if a≠baba\neq b, then ccc is necessarily incident with ppp. Otherwise, a=⟨p,c⟩=bapcba=\langle p,c\rangle=b. Also, without much trouble, one can show that ddd in the condition must be unique.

Pi≠∅subscriptPiP_{i}\neq\varnothing for all 0≤i≤n0in0\leq i\leq n. In other words, there exists at least one flat of every dimension. To see this, we first observe that P0≠∅subscriptP0P_{0}\neq\varnothing, there is at least one point ppp. With ppp, there is a point qqq such that (p,q)∉IpqI(p,q)\notin I. Therefore, there is a (unique) line ℓnormal-ℓ\ell that is incident with
both ppp and qqq. Continue this way until we reach i=nini=n.

Any flat of dimension iii is incident with at least i+1i1i+1 points.

Every flat is incident with at least one flat of every dimension.
As a result, the space SSS is incident with every flat of every dimension.

Shadow. For any a∈PaPa\in P, define I⁢(a)={b∈P∣(a,b)∈I}Iaconditional-setbPabII(a)=\{b\in P\mid(a,b)\in I\}, I-⁢(a)={b∈I⁢(a)∣t⁢(b)≤t⁢(a)}superscriptIaconditional-setbIatbtaI^{{-}}(a)=\{b\in I(a)\mid t(b)\leq t(a)\}, and I+⁢(a)={b∈I⁢(a)∣t⁢(b)≥t⁢(a)}superscriptIaconditional-setbIatbtaI^{{+}}(a)=\{b\in I(a)\mid t(b)\geq t(a)\}. For specific type kkk, we also define Ik⁢(a)={b∣t⁢(b)=k⁢ and ⁢(b,a)∈I}subscriptIkaconditional-setbtbk and baII_{k}(a)=\{b\mid t(b)=k\mbox{ and }(b,a)\in I\}. When k=0k0k=0, I0⁢(a)subscriptI0aI_{0}(a), the set of all points incident with aaa, is referred to as the shadow of aaa. We have that I0⁢(a)⊆I-⁢(a)⊆I⁢(a)subscriptI0asuperscriptIaIaI_{0}(a)\subseteq I^{{-}}(a)\subseteq I(a). We also have I-⁢(a)∩I+⁢(a)=It⁢(a)⁢(a)={a}superscriptIasuperscriptIasubscriptItaaaI^{{-}}(a)\cap I^{{+}}(a)=I_{{t(a)}}(a)=\{a\}.

Remark. It is possible to show that (a,b)∈IabI(a,b)\in I if and only if I0⁢(a)⊆I0⁢(b)subscriptI0asubscriptI0bI_{0}(a)\subseteq I_{0}(b) or I0⁢(b)⊆I0⁢(a)subscriptI0bsubscriptI0aI_{0}(b)\subseteq I_{0}(a). Furthermore, if I0⁢(a)⊆I0⁢(b)subscriptI0asubscriptI0bI_{0}(a)\subseteq I_{0}(b), then t⁢(a)≤t⁢(b)tatbt(a)\leq t(b). In particular, a=baba=b if and only if I0⁢(a)=I0⁢(b)subscriptI0asubscriptI0bI_{0}(a)=I_{0}(b). From the last remark above, I⁢(S)=PISPI(S)=P, and in particular Pk=Ik⁢(S)⊂I⁢(S)subscriptPksubscriptIkSISP_{k}=I_{k}(S)\subset I(S). We also have for any flat aaa, I0⁢(a)⊆I0⁢(S)subscriptI0asubscriptI0SI_{0}(a)\subseteq I_{0}(S). This says that every singleton subset of I0⁢(S)subscriptI0SI_{0}(S) is of the form I0⁢(p)subscriptI0pI_{0}(p) for some ppp. The discussion so far suggests the following simpler, more intuitive, formulation of incidence geometry:

Let AAA be a set. An incidence geometry on AAA is a subset PPP of the power set of AAA such that PPP can be partitioned into n+1n1n+1 finite subsets P0,…,PnsubscriptP0normal-…subscriptPnP_{0},\ldots,P_{n} with the following axioms:

1.

P0subscriptP0P_{0} consists of all singleton subsets of AAA and P0subscriptP0P_{0} is non-empty; elements of P0subscriptP0P_{0} are called points of AAA. Since there is an obvious one-to-one correspondence between AAA and P0subscriptP0P_{0}, we shall follow by convention and call elements of AAA points of AAA instead;

2.

Pn={A}subscriptPnAP_{n}=\{A\}; AAA is called the space;

3.

for every element aaa of PisubscriptPiP_{i}, where i<nini<n, there is a point ppp such that p∉apap\notin a;

4.

for every a∈PiasubscriptPia\in P_{i}, where i<nini<n, and point ppp such that p∉ApAp\notin A, there is a unique b∈Pi+1bsubscriptPi1b\in P_{{i+1}} such that a⊂baba\subset b and p∈bpbp\in b; furthermore, if there is a ccc with a⊂caca\subset c and p∈cpcp\in c, then b⊆cbcb\subseteq c;

5.

for every a∈PiasubscriptPia\in P_{i}, where i>0i0i>0, then there is a point ppp and a b∈Pi-1bsubscriptPi1b\in P_{{i-1}} with p∉bpbp\notin b, such that p∈apap\in a and b⊂abab\subset a;

6.

if a,b∈PiabsubscriptPia,b\in P_{i} and d∈Pi+1dsubscriptPi1d\in P_{{i+1}}, where 0<i<n0in0<i<n, with a point ppp such that p∈a⊂dpadp\in a\subset d and p∈b⊂dpbdp\in b\subset d, then there is a c∈Pi-1csubscriptPi1c\in P_{{i-1}} such that c⊂acac\subset a and c⊂bcbc\subset b.

If we define III on PPP to be (a,b)∈IabI(a,b)\in I if and only if there is a symmetrized inclusion relation between aaa and bbb (a⊆baba\subseteq b or b⊆abab\subseteq a), it is not hard to verify that III is an incidence relation on PPP.

Remarks. Elements of P1subscriptP1P_{1} are called lines of AAA and elements of P2subscriptP2P_{2} are called planes of AAA. Whenever a,b∈PabPa,b\in P such that a⊂baba\subset b, then we say that aaa lies on bbb or bbb passes through aaa. Two special types of incidence geometries are worth mentioning:

If n=2n2n=2, an incidence geometry on AAA is called a plane incidence geometry. In a plane incidence geometry, Axioms 1 through 3, 5 and first part of 4 are necessary. Axiom 2 says that the space is the unique plane of AAA. Axiom 3 enumerates elements of PPP. Axiom 4 is the heart of the incidence geometry; it
says that two distinct points lie on a unique line. Furthermore, Axiom 4, together with Axiom 3, say that any line is a subset of the plane. Second part of Axiom 4 is redundant in a plane incidence geometry. If any element of PPP that passes through two distinct points must be either a line or the plane. If it is a line, it must be the unique line determined by the two points, or the plane, which, clearly includes the unique line. Axiom 5 says that there is
only to create lines (and the plane), namely, via Axiom 4. Axiom 6 is trivial too (let c=pcpc=p).

If n=3n3n=3, an incidence geometry on AAA is called a solid incidence geometry. Axioms 1 through 3, 5, and the first part of Axiom 4 here play the same role as they do in a plane incidence geometry. First part of Axiom 4 also says that a line and a point not lying on it determine a unique plane. The second part of Axiom 4 and Axiom 6 play an equally important role as the other Axioms. Without the second part of Axiom 4, we would not be able to show, for example, that given a plane ππ\pi and a point ppp lying on ππ\pi, there is a line ℓnormal-ℓ\ell lying on ππ\pi but not passing throughppp. Axiom 6 is decidedly non-trivial in solid incidence geometry. It basically says that two planes passing through a common point must pass through a line. Without it, it is possible to find an example such that two planes “intersect” at exactly one point.

Several familiar concepts concerning particular incidence properties of flat can be defined: points are collinear if they lie on the same line; points and lines are coplanar if they lie on the same plane; a pencil is a collection of flats of the same dimension sharing a common incidence property which, in most cases, states they have the same “intersection”.

Speaking of intersections, it would be proper to formally define what it means for two hyperplanes to “intersect”.

Intersection. Let a,b∈PabPa,b\in P. An intersection of aaa and bbb is a flat ccc, if it exists, such that I0⁢(c)=I0⁢(a)∩I0⁢(b)subscriptI0csubscriptI0asubscriptI0bI_{0}(c)=I_{0}(a)\cap I_{0}(b).

Immediately, we see that, if an intersection of aaa and bbb exists, it must be unique. For if I0⁢(c)=I0⁢(a)∩I0⁢(b)=I0⁢(d)subscriptI0csubscriptI0asubscriptI0bsubscriptI0dI_{0}(c)=I_{0}(a)\cap I_{0}(b)=I_{0}(d), then c=dcdc=d. We shall abuse the use of set-theoretic intersection to mean incidental intersection: if aaa and bbb are two flats, then a∩baba\cap b denotes their intersection. Furthermore, if no intersection exists, we write a∩b=∅aba\cap b=\varnothing.

Remarks.

It is easy to show that if aaa and bbb be flats with i=t⁢(a)≤t⁢(b)itatbi=t(a)\leq t(b) and a∩b=d≠∅abda\cap b=d\neq\varnothing, then (d,a)∈IdaI(d,a)\in I and (d,b)∈IdbI(d,b)\in I. In addition, if (a,b)∈IabI(a,b)\in I, then a=dada=d. Also, the unique ccc in Condition 6 above is the intersection or aaa and bbb.

In light of the introduction of the concept of the intersection (of two flats), it seems feasible to toss in an additional element, called the empty block or empty flat, ∅\varnothing, into the underlying set PPP of the incidence geometry: P-1:={∅}assignsubscriptP1P_{{-1}}:=\{\varnothing\} and P′=P-1∪PsuperscriptPnormal-′subscriptP1PP^{{\prime}}=P_{{-1}}\cup P. If we next define a binary relationI′superscriptInormal-′I^{{\prime}} on P′superscriptPnormal-′P^{{\prime}} to be:

then I′superscriptInormal-′I^{{\prime}} becomes an incidence relation on P′superscriptPnormal-′P^{{\prime}}
if we restrict flat aaa in Condition 3 to be non-empty only. Furthermore, P′superscriptPnormal-′P^{{\prime}}, together with I′superscriptInormal-′I^{{\prime}} have almost all the ingredients of being an incidence geometry, except that the range of the type function has now been extended to include -11-1.

For every pair of non-empty a,b∈P′absuperscriptPnormal-′a,b\in P^{{\prime}}, U=I+⁢(a)∩I+⁢(b)UsuperscriptIasuperscriptIbU=I^{{+}}(a)\cap I^{{+}}(b) is a non-empty set since the space SSS is in it. In addition, since nnn is finite, UUU has a minimal elementccc if we order its elements by their corresponding type numbers. Moreover, ccc is unique. We denote this ccc by ⟨a,b⟩ab\langle a,b\rangle. This definition is consistent with our earlier definition of ⟨⋅,⋅⟩fragmentsnormal-⟨normal-⋅normal-,normal-⋅normal-⟩\langle\cdot,\cdot\rangle when the first coordinate is a point and the second coordinate is a flat not passing through the point.

Collecting all the data above, it is now easy to see that P′superscriptPnormal-′P^{{\prime}}, together with the intersection operator∩\cap, and the angle bracket operator ⟨⋅,⋅⟩fragmentsnormal-⟨normal-⋅normal-,normal-⋅normal-⟩\langle\cdot,\cdot\rangle form a semimodular lattice, if we set a∧b:=a∩bassignababa\wedge b:=a\cap b and a∨b:=⟨a,b⟩assignababa\vee b:=\langle a,b\rangle.

We write a∥bfragmentsaparallel-toba\parallel b. Note that if a∩b=aabaa\cap b=a, then a=baba=b, since t⁢(b)=t⁢(a)tbtat(b)=t(a). So if aaa is parallel to bbb, bbb is parallel to aaa, and we may say that aaa and bbb are parallel. Parallelism is a reflexive and symmetric relation. However, it is not transitive (as in the case of a hyperbolic geometry). Condition 6 above can now be restated as: if two flats of dimensions iii, both lying in a flat of dimension i+1i1i+1, are not parallel, then their intersection is a flat of dimension i-1i1i-1.

An incidence geometry with the condition (or axiom) that every pair
of (non-empty) flats of dimensions >0absent0>0 have non-empty intersection is called a projective incidence geometry.

An incidence geometry with the condition (or axiom) that for every (non-empty) flat aaa of dimension iii with i<nini<n, and any point ppp not lying on aaa, there is a flat bbb passing through ppp, such that a∥bfragmentsaparallel-toba\parallel b, is called an affine incidence geometry. The condition just stated is known as the Playfair’s Axiom.

Note to reader: the historical background of this entry is weak. Any additional historical information on this is welcome!